Bi-center conditions and bifurcation of limit cycles in a class of $Z_{2}$-equivariant cubic switching systems with two nilpotent points

In recent years, there have been extensive studies on the qualitative behaviours of nonsmooth dynamical systems, associated with the vector fields which are either discontinuous or nondifferentiable. The main reason of the increasing interest in these studies is that many practical systems contain nonlinearity and nonsmoothness in their equations and display rich complex dynamical phenomena, as observed in biology [40], nonlinear oscillation [7, 29], dry friction of mechanical engineering [5, 30], power electronics [3] and so on. In fact, fundamental mathematical theory for such nonsmooth systems was established several decades ago, see [1, 41].

Switching system is one important class of nonsmooth systems, which is divided into several regions with different definitions of smooth vector fields. In this paper, we consider the switching systems described in the form of

where $f^{\pm}(x,y)$, $g^{\pm}(x,y)$ are analytic functions, and the boundary $\Sigma$ between these two regions $\Sigma^{\pm}$ defines the switching manifold. Although the vector field could be neither continuous nor differentiable on the boundary $\Sigma$, we recall the Filippov convention [18] that the vector field may be crossing, sliding on or escaping from the boundary. Before presenting our main results, we give an overview on two main problems in the qualitative theory of planar differential vector fields, namely, the center problem and the cyclicity problem.

We recall that an isolated singular point is monodromy in a planar vector field if all nearby orbits rotate about this point. It is well known [2] that a monodromic singular point can only be a center or a focus. The center problem is to determine whether a monodromic critical point is a center or a focus, see [20]. But this problem becomes extremely complicated in the switching systems described by (1). To overcome the difficulty, the authors of [21] presented a useful approach for computing the Lyapunov constants of switching systems, which can be used to derive the linear type center conditions for an elementary singular point characterized by a pair of purely imaginary eigenvalues. This type of centers can be determined by vanishing all the Lyapunov constants. In [9, 10, 24] a complete classification with conditions is given to determine if a singular point is a linear type center for several classes of switching systems. Nevertheless, for a given particular class of polynomial systems, finding a complete solution of the center problem is still extremely difficult.

For a given family of differential systems, the Lyapunov constants needed to solve the center problem are also related to the so called cyclicity problem, i.e., finding the maximum number of limit cycles around a singular point. The investigation on bifurcation of limit cycles in switching systems started a half century ago. In [26], Han and Zhang presented an interesting example to show that limit cycles can bifurcate in switching linear systems from a pseudo-focus, i.e., a focus of either focus-focus, parabolic-focus or parabolic-parabolic type, which cannot occur in smooth linear systems. Later, Freire et al. [19], Huan and Yang [28] independently proved that switching linear systems can have at least $3$ limit cycles. Bastos et al. [4] proved that the number of crossing limit cycles in switching linear systems is at least $7$, which has two regions separated by a cubic curve. Tian and Yu [42] constructed a special class of switching Bautin systems to show the existence of $10$ limit cycles in the neighborhood of a center. Recently, by using the averaging method, Braun et al. [6] obtained a new lower bound with 12 limit cycles bifurcating in quadratic switching systems. On the other hand, by the so-called pseudo-Hopf bifurcation analysis, one more limit cycle can be generated by a proper perturbation. However, not many works have been focused on the studies of cubic switching systems. Yu et al. [46] proved that cubic switching systems can have at least 18 limit cycles around two elementary foci. Chen et al. [12] presented a method to study the cyclicity problem in the neighborhood of the origin and infinity for a class of cubic switching systems. The maximal number of limit cycles obtained so far for cubic switching systems by perturbing an elementary center is $26$, see [23, 38].

The aim of this paper is to study the center and cyclicity problems associated with nilpotent singular points in switching systems. Let us recall some known results about nilpotent singular points. An isolated singular point of a planar polynomial system is said to be a nilpotent singular point if both eigenvalues of the Jacobian matrix of the system evaluated at this singular point are zero but the Jacobian matrix is not identically null. The authors of [14, 22, 31, 44, 39, 45] developed some computationally efficient methods for studying the nilpotent center problem of smooth systems. García [20] presented a technique for bounding the maximal number of limit cycles bifurcating from a family of nilpotent singular points in some symmetric smooth polynomial systems. Recently, Li et al. [32] obtained the conditions on two nilpotent singular points to be centers in a class of cubic smooth systems, and proved that $4$ limit cycles exist around each of the two nilpotent singular points, with $4$ more limit cycles bifurcating from two elementary second-order foci which are generated by breaking the first-order nilpotent foci, leading to a total of $12$ limit cycles, which is up to now the best result for cubic systems with nilpotent singular points.

However, the center problem and bifurcation of small-amplitude limit cycles for nilpotent singular points in switching systems become more challenging than that in the case of elementary centers. Facing the challenging, we generalize the Poincaré-Lyapunov method to compute what will be called generalized Lyapunov constants for switching nilpotent systems, and apply the method to study the bi-center problem and the cyclicity problem for a class of $Z_{2}$-equivariant cubic switching systems with two symmetric nilpotent singular points. Here, we say that a $Z_{q}$-equivariant differential system, whose global phase portraits are invariant under a rotation of $2\pi/q$ ($q\in\mathrm{Z^{+}}$) radians around the origin, has a bi-center at the singular points $e_{1}$ and $e_{2}$ if both the two singular points are centers, which will be called bi-center problem. The study of $Z_{q}$-equivariant polynomial systems is closely related to the well-known Hilbert’s 16th problem, for more details see [13, 33, 34, 36, 37].

Without loss of generality, the $Z_{2}$-equivariant cubic switching systems, based on (1) which satisfies (with the switching manifold $\Sigma$ defined as the $x$-axis)

can be written in the form of the differential equations,

where the $x$-axis is the switching manifold and all parameters are real. As is known, the type of nilpotent singular points can generate much more rich dynamics than that of the elementary one, such that the center of system (2) in the switching manifold can be made up of two center-focus (a center or a focus), or one cusp and one center-focus, or two cusps. In this paper, we assume that system (2) has two nilpotent singular points located at $(\pm 1,0)$, and derive the conditions which ensure the nilpotent singular points $(\pm 1,0)$ of (2) to be bi-center for two cases of critical points, namely, the third-order critical point and the second-order critical point. Moreover, we apply general perturbations to prove the existence of at least $20$ small-amplitude limit cycles bifurcating from the nilpotent bi-center $(\pm 1,0)$, two of them are obtained from a symmetric pseudo-Hopf bifurcation by adding a suitable additional perturbation term. This is a new lower bound (increased from $12$ to $20$) on the number of limit cycles bifurcating in such cubic switching systems associated with nilpotent singular points.

The rest of the paper is organized as follows. In Section 2, we will simplify system (2) for the convenience of analysis and present our main results. In Section 3, we present some formulas which are needed in Sections 4 and 5 for proving the two main theorems. Section 4 is devoted to derive $6$ conditions for $(\pm 1,0)$ of system (2) to be bi-center. Then, in Section 5 we use the $6$ center conditions to construct perturbed systems of (2) to show the bifurcation of $20$ limit cycles from the nilpotent singular points $(\pm 1,0)$. Finally, conclusion with discussion on future work is given in Section 6.

Assuming that system (2) has two singular points at $(\pm 1,0)$ yields that

The Jacobian matrices of (2) evaluated at $(\pm 1,0)$ are given by

It follows from $\mathcal{J}^{+}=\mathcal{J}^{-}$ that

To have $(\pm 1,0)$ being isolated nilpotent singular points, the necessary and sufficient conditions are ${\rm Tr}(\mathcal{J}^{\pm})=\det(\mathcal{J}^{\pm})=0$, but $J^{\pm}$ is not identically zero. It is easy to obtain that $a_{30}=b_{01}+b_{21}=0$, and $b_{30}(a_{01}+a_{21})=0$ (but not $b_{30}=a_{01}+a_{21}=0$), which gives either $b_{30}\neq 0,\,a_{01}+a_{21}=0$ or $b_{30}=0,\,a_{01}+a_{21}\neq 0$, that is

However, when $b_{30}=0$ and $a_{01}+a_{21}\neq 0$, system (2) becomes

It is easy to note that the polynomial equations in (7) have a common factor $y$, and so $(\pm 1,0)$ are not isolated singular points. Thus, $b_{30}\neq 0$.

Further, to make $(\pm 1,0)$ be isolated nilpotent singular points of system (2), we set

which leads to $b_{30}=\frac{1}{2}$. With the above results, system (7) is reduced to

For the sake of convenience, we call the system in (9) for $y>0$ “the upper system” and that for $y<0$ “the lower system”.

Now, we need to discuss the multiplicity of the nilpotent singular points $(\pm 1,0)$ for the upper and the lower systems in (9). In fact, by the symmetry of system (9), we only need to analyze the singular point $(1,0)$. Introducing the transformation $x\rightarrow x+1$ into system (9) results in

and thus the singular point $(1,0)$ of system (9) is shifted to the origin of system (10). Assume that

are the unique solutions of the implicit function equations $\Phi^{\pm}(x,y)+x=0$. Further, we denote that

where

For planar smooth systems, if $k$ is the smallest integer satisfying $\mathfrak{f}_{k}\neq 0$, then the multiplicity of the nilpotent singular point is exactly $k$, for more details see [32]. Thus, we can use Theorem 3.5 in [16] to determine the type of the nilpotent singular point $(0,0)$ of (10) for the smooth polynomial equations either in the upper system or in the lower system.

More precisely, when $\mathfrak{g}_{n}=0$ and $\mathfrak{f}_{m}\neq 0$, we have that

When $\mathfrak{g}_{n}\neq 0$, $\mathfrak{f}_{m}\neq 0$ and $\Delta=4(n+1)\mathfrak{f}_{m}+\mathfrak{g}_{n}^{2}$, we have that

where H-E denotes the local phase portrait with one singular point consisting of one hyperbolic sector and one elliptic sector.

Next, we derive the conditions for the nilpotent singular points ($\pm 1$,0) of the $Z_{2}$-equivariant cubic switching system (9) to be bi-center, yielding the following result.

Moreover, to find the maximal number of limit cycles around the nilpotent singular points ($\pm 1$,0) of system (9), we construct perturbed $Z_{2}$-equivariant switching systems using the $6$ bi-center conditions and prove that system (9) has at least $20$ limit cycles bifurcating from the nilpotent singular points ($\pm 1$,0). More precisely, we have the following theorem.

In this section, for the convenience of reading, we first briefly describe the Poincaré-Lyapunov method for determining the linear type center in switching polynomial systems divided by the $x$-axis, and then generate this method to study switching systems associated with isolated nilpotent singular points. As described in [10], we consider the following switching system,

where $\delta,a_{ij}^{\pm},b_{ij}^{\pm}\in R$, $\lambda^{\pm}>0$. Let $\Lambda^{\pm}=(\delta,a_{ij}^{\pm},b_{ij}^{\pm},\lambda^{\pm})$ represent two parameter vectors. The origin is a common singular point in both upper and lower systems of (16). With the polar coordinate transformation: $x=r\cos\theta$ and $y=r\sin\theta$, system (16) can be rewritten as

where $c_{ij}^{\pm}(a_{ij}^{\pm},b_{ij}^{\pm})$ and $d_{ij}^{\pm}(a_{ij}^{\pm},b_{ij}^{\pm})$ are polynomials in the parameters $a_{ij}^{\pm}$ and $b_{ij}^{\pm}$. We suppose that the solutions for the upper and the lower systems of (17) are respectively obtained as $r^{+}(\varrho,\Lambda^{+},\theta)=\sum\limits_{k\geq 1}v_{k}^{+}(\Lambda^{+},\theta)\varrho^{k}$ and $r^{-}(\varrho,\Lambda^{-},\theta)=\sum\limits_{k\geq 1}v_{k}^{-}(\Lambda^{+},\theta)\varrho^{k}$ satisfying the initial condition $r^{+}(\varrho,\Lambda^{+},0)=r^{-}(\varrho,\Lambda^{-},\pi)=\varrho$. Then, we denote by

and

the upper half-return map $\Upsilon^{+}(\varrho)$ and the lower half-return map $\Upsilon^{-}(\varrho)$, respectively, where $v_{k}^{\pm}$’s are the coefficients in Taylor expansions. Following the procedure in [21], we obtain the displacement function,

where $V_{k}$ is called the $k$th-order Lyapunov constant of system (16). The origin of system (16) is a center if and only if all the Lyapunov constants in (18) vanish, i.e., $d(\varrho)\equiv 0$ for $0<\varrho\ll 1$. If there exists $\chi_{*}\in(\Lambda^{+},\Lambda^{-})$ such that $V_{1}(\chi_{*})=V_{2}(\chi_{*})=\cdots=V_{k}(\chi_{*})=0$ and $V_{k+1}(\chi_{*})\neq 0$, then any perturbations to system (16) can yield at most $k$ limit cycles bifurcating from the origin. Based on Lemma 4 in [42], we give the sufficient conditions for proving the existence of limit cycles bifurcating from the origin of system (16).

It is worth mentioning that a switching polynomial system can have one more small-amplitude limit cycle when the sliding segment changes its stability by adding constant terms, which is called pesudo-Hopf bifurcation, see [15, 18]. With $\delta>0$ and $b\in\mathbb{R}$, we assume that $f^{\pm}(x,y)$ and $g^{+}(x,y)$ are given in (16), but $g^{-}(x,y)$ has the following form

Then, the upper system of (16) still has a singular point at the origin, while the lower system of (16) has a singular point near the origin. System (16) would have a sliding segment on the switching manifold $y=0$ (the $x$-axis). For small enough $b$, the new switching system exhibits a pseudo-Hopf bifurcation at $b=0$, as shown in Figure 1, which can produce one more small-amplitude limit cycle, see more details in [8].

In the above discussions, we have briefly described the Poincaré-Lyapunov method and the pesudo-Hopf bifurcation for switching polynomial systems with linear type centers. Now, we consider the switching polynomial system with a nilpotent critical point at the origin,

where $F_{1}^{\pm}(x,y)$ and $F_{2}^{\pm}(x,y)$ are real analytic functions without constant and linear terms. We will show that the Poincaré-Lyapunov method can be generalized to analyze center problem and bifurcation of limit cycles for the nilpotent singular point.

We give the following example to illustrate our idea, as the normal form of the nilpotent differential system can be simplified into the system (for example, see [25]),

where $i\geq 2$, $g(x)$, $h(x)$ and $q(x,y)$ are analytic functions satisfying $g(0)=h(0)=0$. For the sake of simplicity, we consider a codimension-2 Bogdanov-Takens (B-T) bifurcation of the $Z_{2}$ cubic normal form (22). Then, we obtain the following normal form with unfolding:

where the term $\alpha x+\beta y$ is called unfolding with sufficiently small parameters $\alpha$ and $\beta$. Note that the origin of system (23) is a linear type singular point when $\alpha<0$ and $\beta=0$. When $\beta=0$, $\alpha\rightarrow 0^{-}$, the linear type origin becomes a nilpotent singular point.

We remark that the linear type origin of system (23) is a center if and only if $d_{3}=0$, and that the nilpotent monodromic origin (when $c_{3}<0$) of system (23) is also a center when $d_{3}=0$. Hence, the main idea of our method is to perturb system (21) and to establish a relation between the unperturbed system and the perturbed system based on the B-T bifurcation theory, which will generate perturbed Lyapunov constants.

To achieve this, we consider the following perturbed system of (21),

where $-\,\varepsilon^{2}y$ is called unfolding with sufficiently small $|\varepsilon|\!\ll\!1$, $G_{1}^{\pm}(x,y,\varepsilon)$ and $G_{2}^{\pm}(x,y,\varepsilon)$ are real polynomials.

It should be noted that the $\varepsilon$-perturbation terms in (24) are applied to the whole system, not restricted to the local behavior. We use the $\varepsilon^{2}$ rather than $\varepsilon$ in the linear perturbation term to avoid the $\varepsilon^{-i}$ ($i>0$) and $\sqrt{|\varepsilon|}$ terms generalted in the later transformed systems. Based on B-T bifurcation theory and the relation established above for the two systems (21) and (24), we know that a nilpotent center can be $\varepsilon$-approximated by a linear type center. More detailed discussions on this subject can be found in [11].

For this type system of (24), we can apply the Poincaré-Lyapunov method and compute the generalized Lyapunov constants $V_{k}(\varepsilon)$. As a matter of fact, we have the displacement function of system (24), given by

in which $V_{ij}$ denotes the $i$th $\varepsilon^{j}$-order Lyapunov constant. We can determine the center conditions for system (24) by vanishing the $\varepsilon$ terms in these generalized Lyapunov constants, thus leading to a set of algebraic conditions which characterize the existence of a nilpotent center of system (21).

Further, we can use the above generalized Poincaré-Lyapunov method to study the bifurcation of limit cycles from the nilpotent center of the switching system (21). By B-T bifurcation theory, we add a linear perturbation term $-\varepsilon^{2}y$ to system (21) such that the origin of this switching system becomes a linear-type center. Then, we compute the generalized Lyapunov constants of the perturbed system with additional all $\varepsilon^{k}$-order perturbation terms. Following the procedure in [17], we can obtain the bifurcation of limit cycles from the nilpotent center as many as possible. Finally, we obtain one more limit cycle in the switching system by considering the pseudo-Hopf bifurcation.

Since the multiplicity of monodromic critical point for smooth nilpotent systems is $2k+1\geq 3$ (see [16]), we know that the smallest multiplicity for a singular point is $3$ if it is a nilpotent focus or center. For the sake of convenience, we call the singular point with multiplicity $3$ the 3rd-order critical point. However, the multiplicity of the monodromic singular point in switching systems can be $2$, i.e., a 2nd-order critical point. Hence, we consider four cases in the following two subsections.